The number of real solutions of equation `2^(x/2)+(sqrt2+1)^x=(5+2sqrt2)^(x/2)` is

Options

A. one
B. two
C. four
D. infinite

Answer

To find the number of real solutions for the equation

\[ 2^{(x/2)} + (\sqrt{2} + 1)^x = (5 + 2\sqrt{2})^{(x/2)}, \]

we will follow these steps:

### Step 1: Rewrite the equation
We can rewrite the equation as:

\[ 2^{(x/2)} + (\sqrt{2} + 1)^x - (5 + 2\sqrt{2})^{(x/2)} = 0. \]

### Step 2: Substitute variables
Let \( y = 2^{(x/2)} \). Then we can express the other terms in terms of \( y \):

- \( (\sqrt{2} + 1)^x = ((\sqrt{2} + 1)^2)^{(x/2)} = (3 + 2\sqrt{2})^{(x/2)} \)
- \( (5 + 2\sqrt{2})^{(x/2)} \) remains as is.

Thus, we can rewrite the equation as:

\[ y + (3 + 2\sqrt{2})^{(x/2)} - (5 + 2\sqrt{2})^{(x/2)} = 0. \]

### Step 3: Analyze the terms
Now, we need to analyze the behavior of the functions involved. Notice that:

1. \( y = 2^{(x/2)} \) is an increasing function.
2. \( (3 + 2\sqrt{2})^{(x/2)} \) is also an increasing function.
3. \( (5 + 2\sqrt{2})^{(x/2)} \) is an increasing function.

Since all terms are increasing functions of \( x \), the left-hand side of the equation is also an increasing function.

### Step 4: Check for intersections
Since the left-hand side is an increasing function and the right-hand side is also an increasing function, they can intersect at most once.

### Step 5: Find the point of intersection
To find the specific point of intersection, we can set \( x = 2 \):

\[ 2^{(2/2)} + (\sqrt{2} + 1)^2 = (5 + 2\sqrt{2})^{(2/2)}. \]

Calculating each term:

- \( 2^{(2/2)} = 2^1 = 2 \)
- \( (\sqrt{2} + 1)^2 = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2} \)
- \( (5 + 2\sqrt{2})^{(2/2)} = 5 + 2\sqrt{2} \)

Now we can check:

\[ 2 + (3 + 2\sqrt{2}) = 5 + 2\sqrt{2}. \]

This simplifies to:

\[ 5 + 2\sqrt{2} = 5 + 2\sqrt{2}. \]

This confirms that \( x = 2 \) is indeed a solution.

### Conclusion
Since the function is increasing and we found one intersection point, we conclude that there is exactly **one real solution** to the equation.

### Final Answer
The number of real solutions of the equation is **1**.

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